Question: Two cities are building bicycle paths. Charles City has built $5\,\text{km}$ of bicycle paths by the end of the first month, and the total length of the paths doubles each month. Tinsel Town has built $21\,\text{km}$ of bicycle paths by the end of the first month, and the total length of the paths increases by $5\,\text{km}$ per month. At the end of which month does the total length of the bicycle paths in Charles City first exceed the length in Tinsel Town?
Explanation: Notice that Charles City's length grows exponentially while Tinsel Town's length grows linearly. This means Charles City's length is bound to exceed Tinsel Town's length at some point. Let's start calculating the length of the bicycle paths in each place to see when that happens. Month Charles City Tinsel Town (Multiply by $2$ each month.) (Add $5$ each month.) $1$ $5$ $21$ $2$ $10$ $26$ $3$ $20$ $31$ $4$ $40$ $36$ In conclusion, the total length of the bicycle paths in Charles City will first exceed the length in Tinsel Town in month number $4$.